Localization in quantum field theory

Abstract

In non-relativistic quantum mechanics, Born’s principle of localization is as follows: For a single particle, if a wave function [Formula: see text] vanishes outside a spatial region [Formula: see text], it is said to be localized in [Formula: see text]. In particular, if a spatial region [Formula: see text] is disjoint from [Formula: see text], a wave function [Formula: see text] localized in [Formula: see text] is orthogonal to [Formula: see text]. Such a principle of localization does not exist compatibly with relativity and causality in quantum field theory (QFT) (Newton and Wigner) or interacting point particles (Currie, Jordan and Sudarshan). It is replaced by symplectic localization of observables as shown by Brunetti, Guido and Longo, Schroer and others. This localization gives a simple derivation of the spin-statistics theorem and the Unruh effect, and shows how to construct quantum fields for anyons and for massless particles with “continuous” spin. This review outlines the basic principles underlying symplectic localization and shows or mentions its deep implications. In particular, it has the potential to affect relativistic quantum information theory and black hole physics.